The changing role of continuity and discontinuity in the history of philosophy and mathematics
Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie/South African Journal of Science and Technology
| Field | Value | |
| Title | The changing role of continuity and discontinuity in the history of philosophy and mathematics Die wisselende rol van kontinuïteit en diskontinuïteit in die geskiedenis van die filosofie en die wiskunde | |
| Creator | Strauss, Danie F.M. | |
| Description | The aim of this article is to highlight the inevitability of employing discreteness and continuity as primitive (indefinable) modes of explanation in the history of philosophy and mathematics. It embodies the general challenge to account for the coherence of what is unique. Gödel emphasises the coherence of ‘primitive concepts’. Greek philosophy already discovered the spatial whole and/or parts relation with its infinite divisibility. During and after the medieval era philosophers toggled between an atomistic appreciation of the continuum and its opposite, for example found in the thought of Leibniz who postulated his law of continuity (lex continui). The discovery of incommensurability (irrational numbers) by the Greeks caused the first foundational crisis of mathematics, as well as its geometrisation. Leibniz and Newton did not resolve the problems surrounding the limit concept and soon it induced the third foundational crisis of mathematics. It caused Frege and the ‘continuum theoreticians’ to assign priority to the continuum – discreteness is a catastrophe. Recently Smooth Infinitesimal Analysis appreciated what is ‘continuous’ as constituting ‘an unbroken or uninterrupted whole’. Intuitionistic mathematics once more proceeded from an emphasis on the whole and/or parts relation. In spite of alternating attempts to understand continuity exclusively, either in arithmetical or in spatial terms, the history of philosophy and mathematics undeniably confirms that the co-conditioning role of these two modes of explanation remains a constant element in reflections on continuity and discontinuity. (The role of continuity and discontinuity within the disciplines of physics and biology will be discussed in a separate article.) Hierdie artikel wil lig werp op die onvermydelikheid van diskreetheid en kontinuïteit as primitiewe (ondefinieerbare) verklaringswyses in die geskiedenis van die filosofie en die wiskunde. Dit beliggaam die algemene uitdaging om rekenskap van die samehang van iets wat uniek is te gee. Gödel beklemtoon die samehang van ‘primitiewe begrippe’. Die Griekse filosofie het reeds die ruimtelike geheel-dele-relasie – en die oneindige verdeelbaarheid daarvan – ontdek. Gedurende en na die Middeleeue het filosowe wipplank gery tussen ’n atomistiese waardering van die kontinuum en die teenoorgestelde daarvan, wat byvoorbeeld in die denke van Leibniz aangetref word as die gepostuleerde wet van kontinuïteit (lex continui). Die ontdekking van ‘inkommensurabiliteit’ (irrasionale getalle) deur die Grieke het aanleiding gegee tot die eerste grondslagkrisis van die wiskunde en die geometrisering daarvan. Leibniz en Newton kon die probleme rondom die limietbegrip nie besleg nie en spoedig sou dit tot die derde grondslagkrisis van die wiskunde aanleiding gee. Dit het Frege en die ‘kontinuum-teoretici’ daartoe gebring om prioriteit aan die kontinuum te gee – diskreetheid is ’n katastrofe. Onlangs waardeer Smooth Infinitesimal Analysis kontinuïteit as iets wat ’n ‘ongebroke en ononderbroke geheel’ daarstel. Die intuïsionistiese wiskunde het opnuut aangesluit by die klem op die ruimtelike geheel-dele-relasie. Ondanks pogings om kontinuïteit eksklusief aritmeties en ruimtelik te verstaan bevestig die geskiedenis van die filosofie en die wiskunde onmiskenbaar dat die medekondisionerende rol van hierdie twee wyses van verklaring ’n konstante metgesel in die nadenke oor kontinuïteit en diskontinuïteit sou bly. (Die rol van kontinuïteit en diskontinuïteit in die fisika en biologie sal in ’n aparte artikel ondersoek word.) | |
| Publisher | AOSIS | |
| Date | 2017-01-31 | |
| Identifier | 10.4102/satnt.v36i1.1388 | |
| Source | Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie; Vol 36, No 1 (2017); 7 bladsye Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie; Vol 36, No 1 (2017); 7 bladsye 2222-4173 0254-3486 | |
| Language | eng | |
| Relation |
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https://journals.satnt.aosis.co.za/index.php/satnt/article/view/1388/3245
https://journals.satnt.aosis.co.za/index.php/satnt/article/view/1388/3244
https://journals.satnt.aosis.co.za/index.php/satnt/article/view/1388/3246
https://journals.satnt.aosis.co.za/index.php/satnt/article/view/1388/3238
https://journals.satnt.aosis.co.za/index.php/satnt/article/view/1388/3385
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